Properties

Label 80.4.c.c
Level $80$
Weight $4$
Character orbit 80.c
Analytic conductor $4.720$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 80.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.72015280046\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + ( - \beta_{3} - \beta_1 - 1) q^{5} + ( - \beta_{2} + 2 \beta_1) q^{7} + ( - 2 \beta_{3} - \beta_{2} - \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + ( - \beta_{3} - \beta_1 - 1) q^{5} + ( - \beta_{2} + 2 \beta_1) q^{7} + ( - 2 \beta_{3} - \beta_{2} - \beta_1 - 1) q^{9} + ( - 4 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 20) q^{11} + ( - 7 \beta_{2} - 5 \beta_1) q^{13} + ( - 5 \beta_{2} - 10 \beta_1 + 20) q^{15} + (2 \beta_{2} + 18 \beta_1) q^{17} + ( - 4 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 20) q^{19} + ( - 2 \beta_{3} - \beta_{2} - \beta_1 - 68) q^{21} + (21 \beta_{2} + 8 \beta_1) q^{23} + (4 \beta_{3} + 15 \beta_{2} + 9 \beta_1 + 69) q^{25} + ( - 10 \beta_{2} + 8 \beta_1) q^{27} + (8 \beta_{3} + 4 \beta_{2} + 4 \beta_1 + 70) q^{29} + (24 \beta_{3} + 12 \beta_{2} + 12 \beta_1 - 96) q^{31} + ( - 20 \beta_{2} - 56 \beta_1) q^{33} + ( - 4 \beta_{3} - 20 \beta_{2} - 19 \beta_1 + 76) q^{35} + (21 \beta_{2} - 13 \beta_1) q^{37} + (24 \beta_{3} + 12 \beta_{2} + 12 \beta_1 + 56) q^{39} + (6 \beta_{3} + 3 \beta_{2} + 3 \beta_1 - 262) q^{41} + (18 \beta_{2} + 39 \beta_1) q^{43} + (3 \beta_{3} + 15 \beta_{2} + 8 \beta_1 + 193) q^{45} + (11 \beta_{2} - 54 \beta_1) q^{47} + (6 \beta_{3} + 3 \beta_{2} + 3 \beta_1 + 123) q^{49} + ( - 40 \beta_{3} - 20 \beta_{2} - 20 \beta_1 - 480) q^{51} + ( - 9 \beta_{2} + 101 \beta_1) q^{53} + (24 \beta_{3} + 30 \beta_{2} + 34 \beta_1 + 404) q^{55} + ( - 20 \beta_{2} - 16 \beta_1) q^{57} + ( - 44 \beta_{3} - 22 \beta_{2} - 22 \beta_1 + 348) q^{59} + (14 \beta_{3} + 7 \beta_{2} + 7 \beta_1 - 346) q^{61} + ( - 37 \beta_{2} - 32 \beta_1) q^{63} + ( - 28 \beta_{3} - 45 \beta_{2} + 57 \beta_1 + 152) q^{65} + ( - 18 \beta_{2} - 25 \beta_1) q^{67} + ( - 58 \beta_{3} - 29 \beta_{2} - 29 \beta_1 + 28) q^{69} + ( - 16 \beta_{3} - 8 \beta_{2} - 8 \beta_1 - 344) q^{71} + (64 \beta_{2} - 28 \beta_1) q^{73} + ( - 40 \beta_{3} + 85 \beta_1 - 40) q^{75} + ( - 56 \beta_{2} - 100 \beta_1) q^{77} + (8 \beta_{3} + 4 \beta_{2} + 4 \beta_1 - 368) q^{79} + ( - 50 \beta_{3} - 25 \beta_{2} - 25 \beta_1 - 371) q^{81} + ( - 114 \beta_{2} - 65 \beta_1) q^{83} + (8 \beta_{3} - 70 \beta_{2} - 182 \beta_1 + 288) q^{85} + (40 \beta_{2} + 142 \beta_1) q^{87} + (24 \beta_{3} + 12 \beta_{2} + 12 \beta_1 + 330) q^{89} + (80 \beta_{3} + 40 \beta_{2} + 40 \beta_1 - 248) q^{91} + (120 \beta_{2} + 120 \beta_1) q^{93} + ( - 16 \beta_{3} + 30 \beta_{2} - 6 \beta_1 + 364) q^{95} + (46 \beta_{2} + 86 \beta_1) q^{97} + (44 \beta_{3} + 22 \beta_{2} + 22 \beta_1 + 788) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{5} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{5} - 4 q^{9} - 80 q^{11} + 80 q^{15} + 80 q^{19} - 272 q^{21} + 276 q^{25} + 280 q^{29} - 384 q^{31} + 304 q^{35} + 224 q^{39} - 1048 q^{41} + 772 q^{45} + 492 q^{49} - 1920 q^{51} + 1616 q^{55} + 1392 q^{59} - 1384 q^{61} + 608 q^{65} + 112 q^{69} - 1376 q^{71} - 160 q^{75} - 1472 q^{79} - 1484 q^{81} + 1152 q^{85} + 1320 q^{89} - 992 q^{91} + 1456 q^{95} + 3152 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{3} + 2\nu^{2} + 6\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{3} + 6\nu^{2} - 6\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -4\nu^{3} - 4\nu^{2} + 12\nu ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + 2\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{2} + 3\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{3} - 3\beta_{2} + 3\beta_1 ) / 8 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/80\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(21\) \(31\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
1.22474 1.22474i
−1.22474 1.22474i
−1.22474 + 1.22474i
1.22474 + 1.22474i
0 6.89898i 0 −10.7980 + 2.89898i 0 12.6969i 0 −20.5959 0
49.2 0 2.89898i 0 8.79796 + 6.89898i 0 16.6969i 0 18.5959 0
49.3 0 2.89898i 0 8.79796 6.89898i 0 16.6969i 0 18.5959 0
49.4 0 6.89898i 0 −10.7980 2.89898i 0 12.6969i 0 −20.5959 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 80.4.c.c 4
3.b odd 2 1 720.4.f.m 4
4.b odd 2 1 40.4.c.a 4
5.b even 2 1 inner 80.4.c.c 4
5.c odd 4 1 400.4.a.v 2
5.c odd 4 1 400.4.a.x 2
8.b even 2 1 320.4.c.h 4
8.d odd 2 1 320.4.c.g 4
12.b even 2 1 360.4.f.e 4
15.d odd 2 1 720.4.f.m 4
20.d odd 2 1 40.4.c.a 4
20.e even 4 1 200.4.a.k 2
20.e even 4 1 200.4.a.l 2
40.e odd 2 1 320.4.c.g 4
40.f even 2 1 320.4.c.h 4
40.i odd 4 1 1600.4.a.cf 2
40.i odd 4 1 1600.4.a.cm 2
40.k even 4 1 1600.4.a.ce 2
40.k even 4 1 1600.4.a.cl 2
60.h even 2 1 360.4.f.e 4
60.l odd 4 1 1800.4.a.bk 2
60.l odd 4 1 1800.4.a.bp 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.4.c.a 4 4.b odd 2 1
40.4.c.a 4 20.d odd 2 1
80.4.c.c 4 1.a even 1 1 trivial
80.4.c.c 4 5.b even 2 1 inner
200.4.a.k 2 20.e even 4 1
200.4.a.l 2 20.e even 4 1
320.4.c.g 4 8.d odd 2 1
320.4.c.g 4 40.e odd 2 1
320.4.c.h 4 8.b even 2 1
320.4.c.h 4 40.f even 2 1
360.4.f.e 4 12.b even 2 1
360.4.f.e 4 60.h even 2 1
400.4.a.v 2 5.c odd 4 1
400.4.a.x 2 5.c odd 4 1
720.4.f.m 4 3.b odd 2 1
720.4.f.m 4 15.d odd 2 1
1600.4.a.ce 2 40.k even 4 1
1600.4.a.cf 2 40.i odd 4 1
1600.4.a.cl 2 40.k even 4 1
1600.4.a.cm 2 40.i odd 4 1
1800.4.a.bk 2 60.l odd 4 1
1800.4.a.bp 2 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 56T_{3}^{2} + 400 \) acting on \(S_{4}^{\mathrm{new}}(80, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 56T^{2} + 400 \) Copy content Toggle raw display
$5$ \( T^{4} + 4 T^{3} - 130 T^{2} + \cdots + 15625 \) Copy content Toggle raw display
$7$ \( T^{4} + 440 T^{2} + 44944 \) Copy content Toggle raw display
$11$ \( (T^{2} + 40 T - 1136)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 5600 T^{2} + \cdots + 6801664 \) Copy content Toggle raw display
$17$ \( T^{4} + 16896 T^{2} + \cdots + 14745600 \) Copy content Toggle raw display
$19$ \( (T^{2} - 40 T - 1136)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 48440 T^{2} + \cdots + 259467664 \) Copy content Toggle raw display
$29$ \( (T^{2} - 140 T - 1244)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 192 T - 46080)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 75488 T^{2} + \cdots + 314849536 \) Copy content Toggle raw display
$41$ \( (T^{2} + 524 T + 65188)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 90360 T^{2} + \cdots + 576576144 \) Copy content Toggle raw display
$47$ \( T^{4} + 206328 T^{2} + \cdots + 9927332496 \) Copy content Toggle raw display
$53$ \( T^{4} + 624608 T^{2} + \cdots + 72090102016 \) Copy content Toggle raw display
$59$ \( (T^{2} - 696 T - 64752)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 692 T + 100900)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 52280 T^{2} + \cdots + 565868944 \) Copy content Toggle raw display
$71$ \( (T^{2} + 688 T + 93760)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 621440 T^{2} + \cdots + 9130184704 \) Copy content Toggle raw display
$79$ \( (T^{2} + 736 T + 129280)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 1440440 T^{2} + \cdots + 365991120784 \) Copy content Toggle raw display
$89$ \( (T^{2} - 660 T + 53604)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 478208 T^{2} + \cdots + 26342588416 \) Copy content Toggle raw display
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